[math-fun] "the lunatic asylum as my goal"

This summer's AMS Notices reproduces Ramanujan's letter to Hardy where he asserts 1 + 2 + 3 + 4 + ... = - 1/12 On the hypothesis that even the errors of genius may hold value, I've always been curious what his theory was under which this result was derived. Is how he actually got this known? Do you have any conjectures? Or is it just an obvious common fallacy that "everybody knows"? Speculation: it reminds me how the 2-adic expansion familiar from binary computer arithmetic 1 + 2 + 4 + 8 + ... = - 1 can be interpreted by viewing it as the geometric sum y = 1/(1-x) evaluated at x=2. But, for example, dy/dx hasn't seemed fruitful. Can you construct some such way to get Ramanujan's result? Creative lunacy, anyone?